Implications

We have the following indirect implication of form equivalence classes:

43 \(\Rightarrow\) 412
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
43 \(\Rightarrow\) 411 Dependent Choice and Weak Compactness, Delhomme-Morillon-2000[2000], Notre Dame J. Formal Logic
411 \(\Rightarrow\) 412 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

411:

RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.

412:

RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.

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